of the resulting coupling matrix \(J^{(n)}\)  at each time step w.r.t. the Hebbian kernel eq. (REF ) for different values of the ratio \(\tau /\tau ^{\prime }\)  (panel a) and parameters \(\alpha =K/N\)  (panel b), \(N\)  (panel c) and \(\beta \)  (panel d). First, we notice that if the noise is low enough (i.e. \(\beta \gg 1\) ) and if the ratio \(\tau /\tau ^{\prime }\)  is low enough, the dynamics of the coupling matrix converges to the Hebbian kernel. As a matter of fact, there is always a reconstruction error of the Hebbian kernel, in fact, the Frobenius norm of the difference between the synaptic matrix and the Hebbian kernel has an asymptotic value which, in general, is different from zero and depends on \(\tau /\tau ^{\prime }\)  and \(\beta \)  (see panel a,d). In Appendix  we study the fluctuations of the coupling matrix \(J\) , resulting from the Langevin dynamics around the Hebbian fixed point given in (REF ). We find that in the case \(\beta \gg 1\)  the fluctuation is vanishing like \({(\tau /\tau ^{\prime })}^{1/2}\)  (see equation (REF )) and perfectly reproduces the asymptotic behavior of the Frobenius norm of the difference between the synaptic matrix and the corresponding Hebbian kernel (see panel a, dashed line). Whereas, if \(\beta \sim 1\)  we have to take into account also the contribution of the fast noise to the fluctuation around the Hebbian kernel and this is given by a more general formula (see equation (REF )), also in this case we have a perfect agreement between the theoretical estimations and the numerical simulations (panel d, dashed line). Finally, in the panels b and c, we observe that the fluctuation around the Hebbian kernel does depend nor on the size of the system \(N\)  and neither on the parameter \(\alpha =K/N\)  as expected from the theoretical estimation; this implies that the algorithm is robust in performing the storage of the patterns in the synaptic matrix. Moreover, for values of the storage capacity larger than \(\alpha _c \sim 0.14\)  the magnetization does not saturate to one any longer in full agreement with the statistical mechanical limiting description provided by AGS theory [1]}, see Fig. REF .
<FIGURE>